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NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7 - Correlation

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Exercise/Page

NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7 - Correlation Page/Excercise 104

Solution 1

The correct answer is option (iii). If there is no relationship between the two variables, then r = 0. It indicates the non-existence of correlation between height in feet and weight in kilograms.

Solution 2

The correct answer is option (ii). The range of simple correlation coefficient is minus one to plus one (±1). If r = +1, then there is perfect positive correlation. While r = -1, there is negative correlation between the two variables.

Solution 3

The correct answer is option (i). If rxy is positive, then the relation between X and Y is of the type positive correlation, i.e. when Y variable increases, the variable X also increases. In this case, the two variables move in the same direction.

NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7 - Correlation Page/Excercise 105

Solution 4

The correct answer is option (ii). If rxy = 0, the variables X and Y are uncorrelated. There is no linear relation between them. However, they may be related to each other non-linearly.

Solution 5

The correct answer is option (iii). A scatter diagram provides a visual idea about the nature of association between the two variables. It is the simplest method of studying the relationship between two variables without calculating any numerical value. Karl Pearson's coefficient of correlation is to be applied only when the deviation of items is taken from actual means and not from assumed means. Similarly, Spearman's rank correlation can be used for calculating rank correlation only for quantitative data.

Solution 6

The correct answer is option (i). If precisely measured data are available, the simple correlation coefficient is more accurate than the rank correlation coefficient. All the properties of the simple correlation are applicable to Spearman's method. However, it is not as accurate as Karl Pearson's method of correlation because all the information concerning the data is not used. 

Solution 7

Correlation coefficient r and covariance measure the degree of linear relationship between variables X and Y. However, the correlation coefficient is generally preferred to covariance because

  1. r has no unit and is a pure number.
  2. It is independent of origin and scale.
  3. Its value lies between 0 and 1. 

Solution 8

No, the value of correlation coefficient (r) cannot lie outside the -1 and 1 range depending on the type of data. However, if the value of r lies outside this range, then it implies error in the estimation of correlation coefficient.  

Solution 9

No, correlation does not measure causation. This is because it measures the degree and intensity of relationship between two variables. Even a high degree of correlation does not necessarily indicate causation between the variables. So, it is necessary to ensure that the variables for correlation analysis are properly selected to make the analysis more useful.

Solution 10

Rank correlation is more precise than simple correlation coefficient in the following ways:

  1. Measurement of the variables: In some areas, a measuring rod or weighing machine is not used to measure the height and weight of people. So, quality information of people is used to assign rank in terms of height and weight. 
  2. Measurement of qualitative data: Attributes such as intelligence, taste and discipline are difficult to quantify. Hence, ranking may be a better alternative to quantification of qualities.
  3. Presence of extreme values: Correlation coefficient between two variables with extreme values may be different without extreme values. Hence, ranking may be a better alternative to simple correlation. 

Solution 11

No, zero correlation does not imply independence of two variables. If there is zero correlation, it means the two variables (say X and Y) are not correlated and there is no linear relation between them. However, some other type of relation exists between them. 

Solution 12

No, simple correlation coefficient cannot measure any type of relationship because it only specifies the magnitude and direction of correlation, i.e. whether correlation is positive or negative. It only indicates that there is non-linear relationship between two variables but does not measure it such as quadratic, trigonometric and cubic.

Solution 13

Please note that you would need to perform this activity and interpret the result accordingly.

Solution 14

Calculation of coefficient of two variables:

Height of Classmate

X

Height of Benchmate

Y

58

62

69

66

65

58

68

67

71

69

59

70

69

71

 

X-Series

Y-Series

XY

X

X2

Y

Y2

58

3364

62

3844

3596

69

4761

66

4356

4554

65

4225

58

3364

3770

68

4624

67

4489

4556

71

5041

69

4761

4899

59

3481

70

4900

4130

69

4761

61

3721

4209

∑X = 459

∑ X2 = 30257

∑Y = 453

∑Y2 = 29435

∑XY = 29714

 

  

The coefficient of correlation is 0.073. This implies low degree of positive correlation.

Solution 15

List of variables where accurate measurement is difficult:

  1. Qualitative variables such as
    1. Beauty and income earned through modelling  
    2. Honesty and job opportunities in the market 
  2. Subjective variables such as
    1. Poverty and growth of population 
    2. Development and infrastructural facilities

Solution 16

  1. Correlation (r) = 1: If there is perfect positive relationship between two variables, then the value of correlation will be +1.
  2. Correlation (r) = -1: If there is perfect negative relationship between two variables, then the value of correlation will be -1.
  3. Correlation (r) = 0: If there is no relationship between the two variables, then the value of correlation will be zero. However, it does not imply that these two variables are independent. It only indicates non-existence of linear relation between the two variables. 

Solution 17

Rank correlation coefficient differs from Pearsonian correlation coefficient in the following ways:

  1. Karl Pearson's method of correlation measures correlation for quantitative data such as income and savings of the household, while Spearman's method of rank correlation measures correlation for qualitative data such as intelligence and beauty of a person.
  2. Karl Pearson's method of correlation computes deviations from actual or assumed mean, but Spearman's method of rank correlation measures the differences in rank.
  3. Karl Pearson's method of correlation has high significance to extreme values as it is based on actual values, while Spearman's method has low significance to extreme values as it provides them rank. 

Solution 18

 

X-Series

Y-Series

XY

X

X2

Y

Y2

65

4225

67

4489

4355

66

4356

56

3136

3696

57

3249

65

4225

3705

67

4489

68

4624

4556

68

4624

72

5184

4896

69

4761

72

5184

4968

70

4900

69

4761

4830

72

5184

71

5041

5112

∑X = 534

∑X2 = 35788

∑Y = 540

∑Y2 = 36644

∑XY = 36118

 

  

The correlation coefficient is 0.44.

Solution 19

 

X-Series

Y-Series

XY

X 

X2 

Y 

Y2 

-3

9

9

81

-27

-2

4

4

16

-8

-1

1

1

1

-1

1

1

1

1

1

2

4

4

16

8

3

9

9

81

27

∑X = 0

∑X2 = 28

∑Y = 28

∑Y2 = 196

∑XY = 0

 

Coefficient of correlation (r) is 0. This implies that there is no linear relation between X and Y.

NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7 - Correlation Page/Excercise 106

Solution 20

 

X-Series

Y-Series

XY

X 

X2 

Y 

Y2 

1

1

2

4

2

3

9

6

36

18

4

16

8

64

32

5

25

10

100

50

7

49

14

196

98

8

64

16

256

128

∑X = 28

∑X2 = 164

∑Y = 56

∑Y2 = 656

∑XY = 328

 

   

The correlation coefficient between X and Y is 1. Therefore, there is perfect positive relationship between two variables X and Y.

Text Book Solutions

CBSE XI Commerce - Statistics for Economics

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